[Paper Review] The mapping class group of a disk with infinitely many holes

Paul Fabel|ArXiv.org|Mar 4, 2003

Geometric and Algebraic Topology4 references4 citations

TL;DR

This paper constructs a left-orderable, completely metrizable topological group—specifically, the mapping class group of a disk with infinitely many punctures converging to the boundary—that contains Artin’s infinite braid group $B_\infty$ as a dense subgroup under its order topology. The group is realized as the quotient of homeomorphisms fixing the boundary and permuting the punctures, with a metric and left ordering compatible with the topology, resolving a question about completing $B_\infty$ with a group structure.

ABSTRACT

A left orderable completely metrizable topological group is exhibited containing Artin's braid group on infinitely many strands. The group is the mapping class group (rel boundary) of the closed unit disk with a sequence of interior punctures converging to the boundary. This resolves an issue suggested by work of Dehornoy.

Motivation & Objective

• To resolve Dehornoy’s suggestion that a group structure on the completion of $B_\infty$ would be more satisfying than the monoid structure of $EB_\infty$.
• To construct a completely metrizable topological group containing $B_\infty$ as a dense subgroup under its order topology.
• To define a left ordering on the mapping class group of a disk with infinitely many punctures converging to the boundary that is compatible with the quotient topology.
• To establish a complete metric on the mapping class group using the Hausdorff metric and path components of homeomorphisms.

Proposed method

• Define the mapping class group $M(H_\infty)$ as the quotient $H_\infty / H_\infty^0$, where $H_\infty$ is the group of boundary-fixing homeomorphisms of the disk preserving a sequence of punctures converging to the boundary.
• Construct a complete metric $D$ on $M(H_\infty)$ by combining the Hausdorff metric on the orbits of $H_\infty^0$ with the uniform metric on the homeomorphisms.
• Induce a left ordering on $M(H_\infty)$ by lifting Dehornoy’s geometric left ordering from finite braid groups $B_n$ via the smallest index where braid actions differ on connecting arcs.
• Verify compatibility of the order topology with the quotient topology by showing open sets in one topology contain open balls in the other, using uniform control on the metric $D$.
• Use the fact that $H_\infty^0 = \bigcap H_n^0$ is closed in $H_\infty$ to ensure the quotient topology is well-behaved and the metric is complete.
• Leverage the direct limit structure of $B_\infty$ and the monomorphisms $k_n: B_n \to B_{n+1}$ to embed $B_\infty$ densely into $M(H_\infty)$.

Experimental results

Research questions

• RQ1 Can a completely metrizable group be constructed that contains $B_\infty$ as a dense subgroup under its order topology, resolving the need for a group completion of $B_\infty$?
• RQ2 Is there a left ordering on the mapping class group of a disk with infinitely many punctures that is compatible with the quotient topology induced by the path component of the identity?
• RQ3 Does the natural topology on $B_\infty$ induced by Dehornoy’s left ordering admit a completion to a topological group that is both complete and left-orderable?
• RQ4 Can a complete metric be defined on the mapping class group $M(H_\infty)$ using the Hausdorff metric on orbit closures and uniform convergence on the disk?
• RQ5 How does the structure of $H_\infty^0$ as a countable intersection of closed subgroups affect the topology and metric structure of $M(H_\infty)$?

Key findings

• The mapping class group $M(H_\infty)$ is a completely metrizable topological group under the metric $D$ defined via the Hausdorff distance on orbit closures and uniform convergence.
• The group $M(H_\infty)$ admits a left ordering that is compatible with its quotient topology, ensuring the order topology coincides with the metric topology.
• Artin’s infinite braid group $B_\infty$ embeds densely into $M(H_\infty)$ as a subgroup under the order topology, with the embedding being continuous but not an embedding in the topological sense.
• The path component $H_\infty^0$ is closed in $H_\infty$, a nontrivial fact due to the non-locally connected nature of $H_\infty$, which is essential for the metric completeness.
• The metric $D$ is defined by $D(gH_\infty^0, fH_\infty^0) < \varepsilon$ iff there exist representatives $f_1 \in fH_\infty^0$, $g_1 \in gH_\infty^0$, $f_2 \in f^{-1}H_\infty^0$, $g_2 \in g^{-1}H_\infty^0$ such that $\|f_1 - g_1\| < \varepsilon$ and $\|f_2 - g_2\| < \varepsilon$ uniformly on $D^2$, ensuring completeness.
• The order topology on $M(H_\infty)$ is compatible with the quotient topology, as every open set in one topology contains a metric ball from the other, verified via control on the metric $D$ and the behavior on the outer regions $D^2 \setminus D_k$.